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    <title>GROOPS - parametrizationTemporalType</title>

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<h1 id="parametrizationTemporalType">ParametrizationTemporal</h1><p>
This class gives a parametrization of time depending parameters (gravity field, positions, ...).
It will be used to set up the design matrix in a least squares adjustment.
If multiple parametrizations are given the coefficents in the parameter vector
are sequently appended.</p><p>Useally time intervals are defined half open meaning the last time belongs not to the interval.
This behaviour can be changed for the last interval with <strong class="groops-config-element">includeLastTime</strong>.
</p>

<h2 id="constant">Constant</h2><p>
Represents a parameter being constant in time in each <strong class="groops-config-element">interval</strong>.
</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional-unbounded">interval</div></div></td><td><a href="timeSeriesType.html">timeSeries</a></td><td></td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional">includeLastTime</div></div></td><td>boolean</td><td></td></tr>
</table>

<h2 id="trend">Trend</h2><p>
A time variable function is given by a linear trend
\[
f(x,t) = \frac{1}{T}(t-t_0) \cdot f_t(x),
\]with $t_0$ is <strong class="groops-config-element">timeStart</strong> and $T$ is <strong class="groops-config-element">timeStep</strong> in days.
A constant term is not included and must added separately.
</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">timeStart</div></div></td><td>time</td><td>reference time</td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">timeStep</div></div></td><td>time</td><td></td></tr>
</table>

<h2 id="splines">Splines</h2><p>
A time variable function is given by
\[
f(x,t) =  \sum_i f_i(x)\Psi_i(t),
\]with the (spatial) coefficients $f_i(x)$ as parameters and the temporal basis functions $\Psi_i(t)$.
Basis splines are defined as polynomials of degree $n$ in intervals between nodal points in time $t_i$,
for details see <a class="groops-ref" href="fundamentals.basisSplines.html">basis splines</a>.</p><p>The parameters are ordered timewise. First all parameters of $f_{i=1}(x)$ then
$f_{i=2}(x)$ and so on. The total parameter count in each <strong class="groops-config-element">interval</strong> is $N=N_t+d-1$,
where $N_t$ is the count of time points from <a class="groops-class" href="timeSeriesType.html">timeSeries</a> in each interval and $d$
is the <strong class="groops-config-element">degree</strong>.
</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">degree</div></div></td><td>uint</td><td>degree of splines</td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset-unbounded">timeSeries</div></div></td><td><a href="timeSeriesType.html">timeSeries</a></td><td>nodal points in time domain</td></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional-unbounded">intervals</div></div></td><td><a href="timeSeriesType.html">timeSeries</a></td><td></td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional">includeLastTime</div></div></td><td>boolean</td><td></td></tr>
</table>

<h2>Polynomial</h2><p>
A time variable function is represented by Legendre polynomials in each <strong class="groops-config-element">interval</strong>.
The time is normed to $[-1,1)$ in each interval.</p><p>The total parameter count is $(N+1)M$,
where $N$ is the polynmial degree and $M$ the number of intervals.
</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">polynomialDegree</div></div></td><td>uint</td><td>polynomial degree</td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional-unbounded">interval</div></div></td><td><a href="timeSeriesType.html">timeSeries</a></td><td>intervals of polynomials</td></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional">includeLastTime</div></div></td><td>boolean</td><td></td></tr>
</table>

<h2>Oscillation</h2><p>
A time variable function is given by a oscillation
\[
f(x,t) = f^c(\M x)\cos(\omega_i(t)) + f^s(\M x)\sin(\omega_i(t))
\]with $\omega_i=\frac{2\pi}{T_i}(t-t_0)$,
$t_0$ is <strong class="groops-config-element">timeStart</strong> and $T$ is <strong class="groops-config-element">timePeriod</strong> in days.
</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset-unbounded">period</div></div></td><td>time</td><td>[day]</td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">time0</div></div></td><td>time</td><td>reference time</td></tr>
</table>

<h2>Fourier</h2><p>
A time variable function is given by a fourier expansion
\[
f(x,t) = \sum_{m=1}^M f_m^c(\M x)\cos(2\pi m \tau) + f_m^s(\M x)\sin(2\pi m \tau)
\]with the normalized time
\[
\tau = \frac{t-t_A}{t_B-t_A},
\]and $t_A$ is <strong class="groops-config-element">timeStart</strong>, $t_B$ is <strong class="groops-config-element">timeEnd</strong> in each <strong class="groops-config-element">interval</strong>
and $M$ is the <strong class="groops-config-element">fourierDegree</strong>.</p><p>The total parameter count is $2MN$, where $N$ is the number of intervals.
The parameters are sorted in following order: $f_1^c, f_1^s, f_2^c, \ldots$.
</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset">fourierDegree</div></div></td><td>uint</td><td></td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional-unbounded">interval</div></div></td><td><a href="timeSeriesType.html">timeSeries</a></td><td>intervals of fourier series</td></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional">includeLastTime</div></div></td><td>boolean</td><td></td></tr>
</table>

<h2>DoodsonHarmonic</h2><p>
The time variable function is given by a fourier expansion
\[
  f(x,t) = \sum_{i} f_i^c(x)\cos(\Theta_i(t)) + f_i^s(x)\sin(\Theta_i(t)),
\]where $\Theta_i(t)$ are the arguments of the tide constituents $i$
\[
  \Theta_i(t) = \sum_{k=1}^6 n_i^k\beta_k(t),
\]where $\beta_k(t)$ are the Doodson's fundamental arguments ($\tau,s,h,p,N',p_s$) and $n_i^k$
are the Doodson multipliers for the term at frequency $i$.
The multipliers must be given by <a class="groops-class" href="doodson.html">doodson</a> coded as Doodson number
(e.g. 255.555) or as names intoduced by Darwin (e.g. M2).</p><p>The major constituents given by <a class="groops-class" href="doodson.html">doodson</a> can be used to interpolate minor tidal constituents
using the file <a class="groops-class" href="fileFormat_admittance.html">inputfileAdmittance</a>. This file can be created with
<a class="groops-program" href="DoodsonHarmonicsCalculateAdmittance.html">DoodsonHarmonicsCalculateAdmittance</a>.</p><p>The total parameter count is $2N$ with $N$ the number of doodson frequencies.
The parameters are sorted in following order: $f_1^c, f_1^s, f_2^c, \ldots$.
</p>
<table class="table table-hover">
<tr class="table-primary"><th>Name</th><th>Type</th><th>Annotation</th></tr>
<tr class="table-light"><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config mustset-unbounded">doodson</div></div></td><td><a href="doodson.html">doodson</a></td><td>code number (e.g. 255.555) or darwin name (e.g. M2)</td></tr>
<tr class=""><td class="m-0"><div class="h-100 config-tree depth-0"><div class="h-100 config optional">inputfileAdmittance</div></div></td><td>filename</td><td>interpolation of minor constituents</td></tr>
</table>

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